In what follows, OAM modes with negative topological charge will be indicated by placing an explicit minus sign. Now, \(\beta _{-l,m}=\beta _{l,m}\) , a consequence of degeneracy between \(OAM_{-l,m}\)  and \(OAM_{l,m}\)  modes; this follows from the \(l^2\)  dependence of the wave equation [1]}. Furthermore, \(\delta H_{(-l,m)(-l^{\prime },n^{\prime })}=\delta H_{(l,m)(l^{\prime },n^{\prime })}\) , which is seen from Eq. 13. Thus, when \(l\)  is replaced by \(-l\)  in Eqs. 14-17, identical right-hand expressions result. In other words, the perturbation series
\(O_{-l,m}^{\prime }=O_{-l,m}+\sum ^{\prime }_{n,k} a_{(-l,m)(n,k)}^{(1)}O_{n,k}+\sum ^{\prime }_{n,k} a_{(-l,m)(n,k)}^{(2)}O_{n,k}+....,\) 
